3.583 \(\int \frac{\sqrt{\frac{a+b x}{c-b x}}}{a+b x} \, dx\)

Optimal. Leaf size=24 \[ \frac{2 \tan ^{-1}\left (\sqrt{\frac{a+b x}{c-b x}}\right )}{b} \]

[Out]

(2*ArcTan[Sqrt[(a + b*x)/(c - b*x)]])/b

_______________________________________________________________________________________

Rubi [A]  time = 0.0988306, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 \tan ^{-1}\left (\sqrt{\frac{a+b x}{c-b x}}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(a + b*x)/(c - b*x)]/(a + b*x),x]

[Out]

(2*ArcTan[Sqrt[(a + b*x)/(c - b*x)]])/b

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.32937, size = 17, normalized size = 0.71 \[ \frac{2 \operatorname{atan}{\left (\sqrt{\frac{a + b x}{- b x + c}} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x+a)/(-b*x+c))**(1/2)/(b*x+a),x)

[Out]

2*atan(sqrt((a + b*x)/(-b*x + c)))/b

_______________________________________________________________________________________

Mathematica [C]  time = 0.094124, size = 80, normalized size = 3.33 \[ \frac{i \sqrt{c-b x} \sqrt{\frac{a+b x}{c-b x}} \log \left (2 \sqrt{a+b x} \sqrt{c-b x}-i (a+2 b x-c)\right )}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(a + b*x)/(c - b*x)]/(a + b*x),x]

[Out]

(I*Sqrt[c - b*x]*Sqrt[(a + b*x)/(c - b*x)]*Log[2*Sqrt[c - b*x]*Sqrt[a + b*x] - I
*(a - c + 2*b*x)])/(b*Sqrt[a + b*x])

_______________________________________________________________________________________

Maple [B]  time = 0.04, size = 85, normalized size = 3.5 \[ -{(bx-c)\arctan \left ({\frac{2\,bx+a-c}{2\,b}\sqrt{{b}^{2}}{\frac{1}{\sqrt{- \left ( bx+a \right ) \left ( bx-c \right ) }}}} \right ) \sqrt{-{\frac{bx+a}{bx-c}}}{\frac{1}{\sqrt{{b}^{2}}}}{\frac{1}{\sqrt{- \left ( bx+a \right ) \left ( bx-c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)/(-b*x+c))^(1/2)/(b*x+a),x)

[Out]

-arctan(1/2*(b^2)^(1/2)/b*(2*b*x+a-c)/(-(b*x+a)*(b*x-c))^(1/2))*(b*x-c)*(-(b*x+a
)/(b*x-c))^(1/2)/(b^2)^(1/2)/(-(b*x+a)*(b*x-c))^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.804992, size = 32, normalized size = 1.33 \[ \frac{2 \, \arctan \left (\sqrt{-\frac{b x + a}{b x - c}}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-(b*x + a)/(b*x - c))/(b*x + a),x, algorithm="maxima")

[Out]

2*arctan(sqrt(-(b*x + a)/(b*x - c)))/b

_______________________________________________________________________________________

Fricas [A]  time = 0.277303, size = 32, normalized size = 1.33 \[ \frac{2 \, \arctan \left (\sqrt{-\frac{b x + a}{b x - c}}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-(b*x + a)/(b*x - c))/(b*x + a),x, algorithm="fricas")

[Out]

2*arctan(sqrt(-(b*x + a)/(b*x - c)))/b

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a + b x}{- b x + c}}}{a + b x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x+a)/(-b*x+c))**(1/2)/(b*x+a),x)

[Out]

Integral(sqrt((a + b*x)/(-b*x + c))/(a + b*x), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.296876, size = 51, normalized size = 2.12 \[ -\frac{\arcsin \left (\frac{2 \, b x + a - c}{a + c}\right ){\rm sign}\left (a b + b c\right ){\rm sign}\left (b x - c\right )}{{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-(b*x + a)/(b*x - c))/(b*x + a),x, algorithm="giac")

[Out]

-arcsin((2*b*x + a - c)/(a + c))*sign(a*b + b*c)*sign(b*x - c)/abs(b)